Update: according to https://man7.org/linux/man-pages/man7/signal-safety.7.html strcmp() actually is async-signal-safe as of POSIX.1-2008 TC2.
On x86 and some other (mostly extinct) architectures that have string instructions, the string functions are usually best implemented using those (you might get a generation where there's a faster way and then microcode catches back up). And specifically (not just?) on x86 there was/is some confusion about who should or would restore some of the flags that control what these do. So you could end up with e.g. a memcpy or some other string instruction being interrupted by a signal handler and then it would continue doing what it did, but in the opposite direction, giving you wrong results or even resulting in buffer overflows (imagine interrupting a 1 MB memcpy that just started and then resuming it in the opposite direction).
Not necessarily. An implementation might choose to e.g. use some kind of cache similar to what the JVM does with interned strings, and then a function like strcmp() might behave badly if it happened to run while that cache was halfway through being rebuilt.
For arbitrary pointers no. But it could special-case e.g. string constants in the source code and/or pointers returned by some intern function (which is also how the JVM does it - for arbitrary strings, even though they're objects, it's always possible that the object has been GCed and another string allocated at the same location).
For example, recently I wanted to call `gettid()` in a signal handler. Which I guessed was just a simple wrapper around the syscall.
However, it seems this can cache the thread ID in thread local storage (can't remember exact details).
I switched to making a syscall instead.